The unique control features of topological stochastic and quantum systems

TL;DR

This paper compares topological features in stochastic and quantum systems, deriving spectral properties and revealing how non-reciprocity and topology influence state localization and robustness.

Contribution

It provides analytical expressions for spectral properties and uncovers unique control mechanisms and a novel topologically emerging state in stochastic systems.

Findings

Non-reciprocity shifts stochastic states away from steady-state.

Topological features cluster states around the steady-state in stochastic systems.

A new topologically emerging state persists across models and dimensions.

Abstract

Topological phases support edge states that can be robust to material deformations and other perturbations. While well-studied in quantum systems, topological phases have also been observed in stochastic and biochemical systems, yet it remains unclear which of their properties remain similar or different from those in quantum systems. In this paper, we derive analytical expressions for the spectral properties of simple quantum and stochastic models on the same lattice to rigorously characterize these complex systems. Intriguingly, we find that non-reciprocity moves states away from the steady-state in stochastic systems while clustering states at zero-energy in quantum systems. In contrast, making the system more topological does the opposite: it clusters more states around the steady-state in stochastic systems but moves states away from the zero-energy state in quantum systems. These results provide control parameters for selection and modulation of different purposes while quantifying the size of gap which protects the longest-lived states. Lastly, we discover a mode unique to stochastic systems that we dub the topologically emerging state, which persists across different models and dimensions, including in the presence of non-equilibrium currents.